Generating adversaries for request-answer games

The k-server conjecture postulates the existence of an algorithm which is k-competitive for all values of k on all metric spaces. We give a procedure which is guaranteed to find a complementary structurean adversary strategy and a metric space such that no algorithm is k-competitive against the adversary strategy and metric space assllrning such a structure exists. That is, we prove the complement of the k-server conjecture is recursively enumerable. In fact, we give a general procedure that can perform the following search for a fairly general subset of request-an.qwer games [3]. For a given c, find a finite adversary strategy against which no algorithm can be better than c-competitive assnrning such an adversary strategy exists. This essentially implies that for these request-answer games, "Is c u >__ c?" is recursively enumerable, where e u is the optimal competitive ratio for request-answer game II. We have used a modified version of our procedure to find a lower bound of 1.85358 for the online load balancing problem on m _> 80 identical machines, improving slightly upon the lower bound of 1.852 by Albers [1]. We can extend our procedure to produce oblivious adversaries for randomized online algorithms if we explicitly bound the number of random bits used by the randomized algorithm. Request-answer games are a class of online problems defined by Ben-David et al. [3] in which each request must be serviced immediately by the online algorithm. Furthermore, while there may be an infinite number of legal requests, there are only a finite number of welldefined answers or responses for serving a request. Our procedure 0nly applies to a subset of request-answer games which satisfy an additional "linearizability" constraint, but we defer a description of this linearizability constraint until after the following definition of an adversary. Fortunately, many problems satisfy this linearizability constraint including many variations of the